Question: Simplify; express your answer in exponential form. Assume $x\neq 0, z\neq 0$. $\dfrac{{(x^{-1}z^{5})^{2}}}{{(x^{-2}z^{5})^{3}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(x^{-1}z^{5})^{2} = (x^{-1})^{2}(z^{5})^{2}}$ On the left, we have ${x^{-1}}$ to the exponent ${2}$ . Now ${-1 \times 2 = -2}$ , so ${(x^{-1})^{2} = x^{-2}}$ Apply the ideas above to simplify the equation. $\dfrac{{(x^{-1}z^{5})^{2}}}{{(x^{-2}z^{5})^{3}}} = \dfrac{{x^{-2}z^{10}}}{{x^{-6}z^{15}}}$ Break up the equation by variable and simplify. $\dfrac{{x^{-2}z^{10}}}{{x^{-6}z^{15}}} = \dfrac{{x^{-2}}}{{x^{-6}}} \cdot \dfrac{{z^{10}}}{{z^{15}}} = x^{{-2} - {(-6)}} \cdot z^{{10} - {15}} = x^{4}z^{-5}$